Playing with the DLC
The DLC pack adds an assorted collection of costumes and songs to the game. However, to utilize the DLC, you'll need to patch the game, which will result in parts of the interface translation disappearing. It is suggested you beat the game at least once before trying this.
Download DLC pack here: http://www.mediafire.com/?r8sz98z3l7t4zc3
Download Eboot.bin here: http://www.mediafire.com/?5fr9nbw2es1984c
Download UMDgen here: http://www.dashhacks.com/downloads/viewdownload/6-psp-downloads/8144-umdgenv400.html
Patching the .iso:
Step 1: MAKE A BACKUP OF YOUR .ISO IMAGE. IF YOU DO STEPS THREE AND FOUR INCORRECTLY, IT WILL RENDER YOUR .ISO USELESS AND YOU'LL NEED TO REDOWNLOAD.
Step 2: Open up the .iso file with UMDgen.
Step 3: Replace the eboot in the SYSDIR folder with the one downloaded (drag and drop)
Step 4: Save the .iso **AS A NEW ISO**
Step 5: Put the .iso back on your PSP.
Inserting the DLC onto your PSP:
Step 7: Extract the NPXP00100.rar
Step 8: Take the extracted NPXP00100 folder and make sure there isn't a folder with the exact name inside that folder. If there is, use the one inside instead for the next step.
Step 9: Drag it to the '../PSP/GAME' folder.

Emulating the game with PPSSPP (Recommended if you don't have a PSP)

If at all possible, it's suggested that you play the game on an actual PSP (PSP-2000 and PSP-3000 are the best models). However, the latest Windows dev builds of PPSSPP can emulate iDOLM@STER SP with few problems.

1. Download and install the latest full version of DirectX: http://www.microsoft.com/en-au/download/details.aspx?id=35

2. Download the latest PPSSPP dev build: http://www.ppsspp.org/downloads.html#devbuilds

3. Extract the 7z file (use http://www.7-zip.org/) and place the ppsspp folder anywhere you like.

4. Run PPSSPPWindows.exe and follow the prompt to download and install Atrac3+ Audio Decoding Support (required for in-game music to work).

5. From the Options menu, disable Buffered Rendering and Vextex Cache, then enable Simple 2x SSAA and MipMapping. Leave the other options as is. I must stress that which options you need will depend heavily on your personal hardware configuration, so feel free to experiment until you find something that works.
6. Check and adjust the controls from the Options menu, then load up your iDOLM@STER SP .iso using the File menu.

7. The game should now play (mostly) fine. The first Yes/No prompt that comes up will be the 'Do you want to create save data' prompt, so select yes.
If you encounter problems, try changing some of the other options to see what happens. Performance will differ depending on your computer's hardware and software configuration, so you may need to tailor the settings to suit you personally if you get any glitches. For example, try disabling MipMapping and Hardware Transform. Don't worry too much about screwing the settings up, you can always change them back or reset to the defaults: Just delete the ppsspp.ini file that's been created in the ppsspp folder (The file will be automatically recreated with the default settings), then follow the instructions again from step 4. Good luck.

Emulating the game with JPCSP (Not Recommended)
These instructions are provided for legacy's sake.
1. Download JPCSP (Preferably 64-bit version) and Java SE Runtime Environment 7
http://buildbot.orphis.net/jpcsp/
http://www.oracle.com/technetwork/java/javase/downloads/java-se-jre-7-download-432155.html
2. Install Sonicstage
http://sonicstage.en.softonic.com/
3. Download game from here and place iso into the umdimages folder within the JPCSP directory
http://www.nyaa.eu/?page=torrentinfo&tid=266966
4. Download these config properties and replace them with the one in your JPCSP directory
http://www.mediafire.com/?qj6kcjuz151271p
5. Launch JPCSP and Load up the iso. Run the game. (Note: It will take more than 10 seconds to load, dependant on your specs)
6. (Optional) If there are some artifacting on the menu screen, go into Configuration (F12), Video, and recheck Disable optimized Vertexinfo and Use Software Rendering (Bottom two settings). Apply the settings, then uncheck these settings again, and apply the settings again. No restart required. This should fix the artifacting.
These are my configurations for Jpcsp v0.6 r2552. My specs are Phenom II x6 @ 4.0 GHz, 550 Ti, 8 GB of memory, ect. I play the game at 60 fps when not streaming it.
(These settings are primarily optimized for my computer, with high quality emulation in mind. It's best to just fiddle around with the different options till you find an optimal configuration rather than leaving the config as is. Don't be afraid to change the settings around until you find it suitable to your likings.)